查看更多>>摘要：In this work, we introduce a new semi-discrete scheme based on the so-called no-flow curves and its numerical analysis for solving initial value problems that involve one-dimensional (1D) scalar hyperbolic conservation laws and of the form, u(t) + H(u)(x) = 0, x is an element of R, t > 0, u(x, 0) = u(0)(x). In addition, we present a two-dimensional (2D) version of the semi-discrete Lagrangian-Eulerian scheme to show that the proposed method can be applied to multidimensional problems. From an improved 1D weak numerical asymptotic analysis, we found that the solutions provided by the novel semi-discrete scheme satisfy a maximum principle property and a Kruzhkov entropy condition. We also highlight the possibility of using the no-flow curves as a new desingularization analysis technique for the construction of computationally stable numerical fluxes in the locally conservative form for nonlinear hyperbolic problems. We provide nontrivial 1D and 2D numerical examples with nonlinear wave interaction to illustrate the effectiveness and capabilities of the proposed approach and verify the theory. According to the results, the scheme handles discontinuous solutions (shocks) with low numerical dissipation quite well and shows a very good resolution of rarefaction waves with no spurious glitch effect in the vicinity of the sonic points. We also consider a test case for nonstrictly hyperbolic conservation laws with a resonance point (coincidence of eigenvalues) modeling three-phase flow in a porous media transport problem. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In the paper, the authors, (1) by constructing a counterexample and utilizing Minkowski's inequality, demonstrate that there existed errors in the proofs of Theorems 1 and 2 in the paper "Mehrez and Agarwal, (2019)"; (2) with the help of an integral identity and by means of Holder's integral inequality, present a new integral inequality of the Hermite-Hadamard type for convex functions. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：A modified weak Galerkin finite element method is studied for nonmonotone quasilinear elliptic problems. Using the contraction mapping theorem, the uniqueness of the solution to the discrete problem is proved. Moreover, optimal order a priori error estimates are established in both a discrete H-1 norm and the L-2 norm. Numerical experiments are conducted to confirm the theoretical results. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller's boundary classification. We compare the frequently used Euler-Maruyama and Milstein methods, two Lie-Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong-Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler-Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler-Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie-Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features. Crown Copyright (C) 2021 Published by Elsevier B.V.
查看更多>>摘要：This study proposes a new methodology to analyze the impacts of host and pathogen dispersal on coinfection dynamics using the example of Escherichia coli O157:H7 in a dairy farm. A multi-strain Susceptible-Infected-Susceptible model is extended to a Reaction-Diffusion (RD) coinfection model of E. coli O157:H7 transmission, which includes intermittent shedding and pathogen growth in the environment. Analysis of the new RD coinfection model consists of existence and stability of constant equilibria, calculation of the basic reproduction number, and existence of traveling and stationary wave solutions. A stationary wave solution of the RD model defines a nonconstant endemic equilibrium. Whereas a traveling wavefront represents an epidemic wave of infection passing through the farm with a specific speed, direction, and amplitude. The numerical simulations of the RD model demonstrate stable traveling and stationary wavefronts of the RD coinfection model. The stationary wavefront connects two constant coexistence equilibria and the traveling wavefront represents the gradual establishment of an endemic coexistence equilibrium in the dairy farm. The significance of this study lies in utilizing the nonlinear wave theory to analyze dynamics of coinfection in a dairy farm both with respect to location and time. Hence, in addition to Turing theory, the nonlinear wave theory can be used to determine the likelihood of acquiring the infection based on the location of cattle at each specific time. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Starting with the simplest case of an analogue square root control-system, we extend the idea to the discrete-time case and ultimately the multivariable discrete-time case. Several new results then emerge from the multivariable work. A nonlinear system is designed to take the square root of an arbitrary square matrix. The matrix can have real or complex values and an analysis of stability is reached by using matrix calculus and nonlinear theory. The root-locus of the multivariable system exhibits straight-line characteristics which has not been observed before and to achieve stability we introduce the concept of a complex step-size (or gain) in a recursive square root algorithm. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient algorithms for determining such adaptive splines, the difficulty with TV regularization is that the solution is generally non-unique, an aspect that is often ignored in practice. In this paper, we present a systematic analysis that results in a complete description of the solution set with a clear distinction between the cases where the solution is unique and those, much more frequent, where it is not. For the latter scenario, we identify the sparsest solutions, i.e., those with the minimum number of knots, and we derive a formula to compute the minimum number of knots based solely on the data points. To achieve this, we first consider the problem of exact interpolation which leads to an easier theoretical analysis. Next, we relax the exact interpolation requirement to a regression setting, and we consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. Based on our theoretical analysis, we propose a simple and fast two-step algorithm, agnostic to uniqueness, to reach a sparsest solution of this penalized problem. (C)& nbsp;2021 The Author(s). Published by Elsevier B.V.& nbsp;
查看更多>>摘要：In this paper, we present a fourth-order Cartesian grid-based boundary integral method (BIM) for a multiple acoustic scattering problem on closely packed obstacles. We reformulate the exterior Helmholtz boundary value problems (BVPs) as a Fredholm boundary integral equation (BIE) of the second kind for some unknown density function. Unlike the traditional boundary integral method, a distinctive feature of our scheme is that we do not require quadratures and direct evaluations of nearly singular, singular or hyper-singular boundary integrals in the solution of BIEs. Instead, we reinterpret boundary integrals as solutions to equivalent simple interface problems in an extended rectangle domain, which can be solved efficiently by a fourth-order finite difference method coupled with numerical corrections, FFT based solution and interpolations. Extensive numerical experiments show that our method is formally high-order accurate, fast convergent and in particular insensitive to complexity of scatterers. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper provides a Nystrom method for the numerical solution of Volterra integral equations whose kernels contain singularities of algebraic type. It is proved that the method is stable and convergent in suitable weighted spaces. An error estimate is also given as well as several numerical tests are presented. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Two fast numerical algorithms are proposed for computing interval vectors containing positive solutions to M-tensor multi-linear systems. The first algorithm involves only two tensor-vector multiplications. The second algorithm is iterative one, and generally gives interval vectors narrower than those by the first algorithm. We also develop two verification algorithms for Perron vectors of a kind of weakly irreducible nonnegative tensors, which we call slightly positive tensors. The first and second algorithms have properties similar to those of the two algorithms for the solutions to the M-tensor systems. We clarify relations between slightly positive tensors and other tensor classes. Numerical results show efficiency of the algorithms. (C) 2021 Elsevier B.V. All rights reserved.
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